<!DOCTYPE html>
<html>
<title>Sheets Tutorial - Group Explorer 3.0 Help</title>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<link rel="stylesheet" href="../w3.css">
<link rel="stylesheet" href="../w3-theme-blue.css">
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.3.0/css/font-awesome.min.css">
<link rel="stylesheet" href="https://use.fontawesome.com/releases/v5.7.2/css/all.css" integrity="sha384-fnmOCqbTlWIlj8LyTjo7mOUStjsKC4pOpQbqyi7RrhN7udi9RwhKkMHpvLbHG9Sr" crossorigin="anonymous">
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.1.1/jquery.min.js" defer></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<script type="text/x-mathjax-config">
    MathJax.Hub.Config( { TeX: { extensions: [ "color.js" ] } } );
</script>
<script>
  // needed by the help scripts
  var base_url = '..';
</script>
  <link href="../w3.css" rel="stylesheet">
  <link href="../w3-theme-blue.css" rel="stylesheet">
  <script src="../ge-citations.js" defer></script>
  <script src="../search/main.js" defer></script>
<style>
img { border: 1px solid black; }
</style>
<body>

<!-- Header -->
<header class="w3-container w3-theme w3-padding" id="myHeader">
  <!--<i onclick="w3_open()" class="fa fa-bars w3-xlarge w3-button w3-theme"></i>-->
  <div class="w3-center">
  <h1 class="w3-xxxlarge">Group Explorer 3.0 Help: Sheets Tutorial</h1>
  </div>
</header>

<hr>

<div class="w3-container">
    <div class="w3-row w3=border">
        <div class="w3-quarter w3-container">
            <div class="w3-padding w3-card w3-bar-block">
                
                <br>
                <form action='search/index.html'>
                    <input name='q' id='mkdocs-search-query' type='text'
                           placeholder='Enter search term'
                           title='Type search term here'/>
                    <input type='submit' value='Search'
                           class='w3-btn w3-theme'/>
                </form>
                <hr>
                

                <h3>Contents</h3>
                <ul>
                
                    <li><a style="text-decoration: none;"
                          href="#sheets">Sheets</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#making-your-own-sheets">Making your own sheets</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#getting-group-explorer-to-make-sheets-for-you">Getting Group Explorer to make sheets for you</a></li>
                
                </ul>

                <hr>

                <h3><a style="text-decoration: none;"
                      href="http://nathancarter.github.io/group-explorer">Group Explorer Home</a></h3>

                <hr>

                <h3>Other help pages</h3>
                
                    <ul>
                    
                    
                        <li><a href=".."
                               style="text-decoration: none;"
                               class=""
                               >Main Help Page</a></li>
                    
                    
                    
                        <li><a href="../welcome/"
                               style="text-decoration: none;"
                               class=""
                               >Welcome</a></li>
                    
                    
                    
                        <li><a href="../gs-index/"
                               style="text-decoration: none;"
                               class=""
                               >Getting Started</a></li>
                    
                    
                    
                    
                    
                        <li><a href="../tu-index/"
                               style="text-decoration: none;"
                               class=""
                               >Tutorials</a></li>
                    
                    
                    
                    
                    
                        <li><a href="../rf-um-index/"
                               style="text-decoration: none;"
                               class=""
                               >User Manual</a></li>
                    
                    
                    
                    
                    
                    
                    </ul>
                

                <hr>

                <h3>Have an idea?</h3>
                <p><i class="fab fa-github"></i>
                   <a href='http://github.com/nathancarter/group-explorer/edit/master/help-src/tu-sheets.md'
                    >Edit this page</a></p>
            </div>
        </div>
        <div class="w3-threequarter w3-container">
            <h2 id="sheets">Sheets</h2>
<p><em>Group Explorer</em> since version 2.0 has been able to open multiple group
visualizations in one document called a &ldquo;sheet,&rdquo; so that they can be
compared and homomorphisms between them created, illustrated, and studied.</p>
<p>This page provides a quick introduction to sheets.  To go directly to the
details of the sheet interface and how to use it, <a href="../rf-um-sheetwindow/">visit this page of the
User Manual</a>.</p>
<p>Let&rsquo;s begin with &ldquo;Making your own sheets,&rdquo; but you can jump down to
&ldquo;<a href="#getting-group-explorer-to-make-sheets-for-you">Getting Group Explorer to make sheets for
you,</a>&rdquo; below, where the
fancier stuff shows up.</p>
<h2 id="making-your-own-sheets">Making your own sheets</h2>
<p>Let&rsquo;s use a sheet to compare two groups of the same order,
<a href="http://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=Z_6.group"><script type="math/tex">\mathbb{Z}_6</script></a>
and
<a href="http://nathancarter.github.io/group-explorer/GroupInfo.html?groupURL=S_3.group"><script type="math/tex">S_3</script></a>.</p>
<ul>
<li>
<p>From the <em>Group Explorer</em> main page, click the Sheet icon on the top
   right hand side of the page.</p>
<p><img alt="Sheet icon" src="../sheet-icon.png" /></p>
<p>This creates a new sheet.</p>
</li>
<li>
<p>In the control window on the right hand side of the sheet, select
   <script type="math/tex">\mathbb{Z}_6</script> from the &ldquo;Group&rdquo; drop-down list and click the &ldquo;Cycle graph&rdquo;
   button above it. You should see a small cycle graph for <script type="math/tex">\mathbb{Z}_6</script>
   appear in the upper left-hand corner of the left side of the sheet. I resized
   mine slightly and the result was as follows. (To move a visualizer just drag
   it with the mouse or your finger; to resize it right-click [tap] it and
   select &lsquo;Resize&rsquo; from the resulting context menu. A light blue outline will
   appear around the visualizer. Then drag starting outside the element in the
   direction you&rsquo;d like it to grow. A short click [tap] will dismiss the blue
   outline. See <a href="../rf-um-sheetwindow/#resizing">here</a> for more details.)</p>
</li>
</ul>
<p><img alt="Sheet containing one cycle graph" src="../illustration-sheettut1.png" /></p>
<ul>
<li>But that&rsquo;s just one visualizer; we want to have at least two.  So let&rsquo;s
   repeat the same steps for inserting a cycle graph for <script type="math/tex">S_3</script> as well, but
   move it a little to the right of the first one, as shown below.</li>
</ul>
<p><img alt="Sheet containing two cycle graphs" src="../illustration-sheettut2.png" /></p>
<ul>
<li>Now let&rsquo;s start comparing these groups.  In group theory, the means of
   examining relationships between groups is via
   <a href="../rf-groupterms/#homomorphism">homomorphisms</a>.  So let&rsquo;s create one in this
   sheet, as follows:<ul>
<li>Right-click [tap] the left cycle graph element to open its context
  menu</li>
<li>Select the &ldquo;Create Map&rdquo; option from menu</li>
<li>Click [tap] the other cycle graph, the homomorphism target.</li>
<li>You should see a new function <script type="math/tex">f</script> appear, connecting the two graphs.</li>
</ul>
</li>
</ul>
<p><img alt="Sheet containing two cycle graphs connected by a homomorphism f" src="../illustration-sheettut3.png" /></p>
<ul>
<li>So far this isn&rsquo;t very informative, but if we right-click [tap] the
   morphism label box to open its context menu and select &ldquo;Edit&rdquo;, we can do all
   sorts of interesting things. For instance, you can decide which elements from
   <script type="math/tex">\mathbb{Z}_6</script> should correspond to which elements from <script type="math/tex">S_3</script>.
   Furthermore, <em>Group Explorer</em> will not let you mess this up (you cannot
   define a non-homomorphism.)  The morphism editing dialog is shown below.</li>
</ul>
<p><img alt="Homomorphism editing dialog" src="../illustration-sheettut4.png" /></p>
<ul>
<li>The homomorphism defaults to the zero map (all elements map to the identity,
   in this case <script type="math/tex">e</script>) but you can change it, of course. I will map <script type="math/tex">a</script> to
   <script type="math/tex">r</script> and then check the &ldquo;Draw multiple arrows&rdquo; box above.  The result is the
   following illustration of one way to map <script type="math/tex">\mathbb{Z}_6</script> to <script type="math/tex">S_3</script>.</li>
</ul>
<p><img alt="Homomorphism from Z_6 to S_3" src="../illustration-sheettut5.png" /></p>
<p>The arrows require some attention to follow carefully, but you can see how the
six-element circle marches around the little three-element circle twice.  I&rsquo;ve
taken the liberty of highlighting <script type="math/tex">\mathbb{Z}_6</script> red and its
<a href="../rf-groupterms/#image-of-a-subset-under-a-morphism">image</a> in <script type="math/tex">S_3</script> red
also. (To do so, open the element context menu for the visualizer you&rsquo;d like to
highligh and select &ldquo;Edit&rdquo;. Then play with its subsets <a href="../rf-um-subsetlistbox/">as documented
here</a>.)</p>
<p><img alt="Homomorphism from Z_6 to S_3 with domain and image highlighted" src="../illustration-sheettut6.png" /></p>
<p>This is only the beginning of the potential of sheets. The next section
shows much more.</p>
<h2 id="getting-group-explorer-to-make-sheets-for-you">Getting <em>Group Explorer</em> to make sheets for you</h2>
<p>The <a href="../rf-um-groupwindow/">group info pages</a> of <em>Group Explorer</em> are full of
links that create sheets. For many common computations, it is very
interesting to be able to see the result of the computation visually. I will
whet your appetite for such illustrations by giving a few examples here, and
providing links for you to browse further yourself.</p>
<ul>
<li>To see a <a href="../rf-groupterms/#short-exact-sequence">short exact sequence</a> exhibiting the
   <a href="../rf-groupterms/#normal-subgroup">normality</a> of a
   <a href="../rf-groupterms/#subgroup">subgroup</a>
   (and the <a href="../rf-groupterms/#quotient-group">quotient group</a> it computes):<ul>
<li>Expand the &ldquo;Subgroups&rdquo; section under &ldquo;Computated Properties&rdquo; in the
  <a href="../rf-um-groupwindow/#computed-properties">group info page</a>.</li>
<li>Then find the subgroup in question on the list and follow the link
  provided.</li>
<li>The illustration below shows the normality of <script type="math/tex">V_4</script> in <script type="math/tex">A_4</script>.</li>
</ul>
</li>
</ul>
<p><img alt="A connection of five groups illustrating the normality of V_4 in A_4" src="../a_4_sheet_ses.png" /></p>
<ul>
<li>To see a <a href="../rf-groupterms/#lattice-of-subgroups">lattice of subgroups</a>
   for a given group:<ul>
<li>Again, expand the &ldquo;Subgroups&rdquo; section of the group info page.</li>
<li>Follow the link provided at the top of the resulting page,
  offering to create a sheet showing the lattice of subgroups.</li>
<li>The illustration below shows all subgroups of <script type="math/tex">S_3</script>.</li>
</ul>
</li>
</ul>
<p><img alt="The lattice of subgroups for S_3" src="../illustration-sheettut8.png" /></p>
<ul>
<li>To see the <a href="../rf-groupterms/#solvable-group-solvable-decomposition">solvable decomposition</a> of any
   <a href="../rf-groupterms/#solvable-group-solvable-decomposition">solvable group</a>:<ul>
<li>Expand the &ldquo;Solvable&rdquo; section of the group info page.</li>
<li>The decomposition will be reported in text and you can click any of
  several links on that page to see it illustrated in various ways.</li>
<li>The illustration below shows the solvable decomposition for <script type="math/tex">S_4</script>.</li>
</ul>
</li>
</ul>
<p><img alt="The solvable decomposition for S_4" src="../illustration-sheettut9.png" /></p>
        </div>
    </div>
</div>

<hr>

<!-- Footer -->
<footer class="w3-container w3-blue w3-padding-16">
  <p>Based on a <a href="https://www.w3schools.com/w3css/default.asp" target="_blank">w3.css</a> template.
  Thanks to them for sharing!</p>
</footer>

</body>
</html>